I'm trying to solve Exercise 8.2.4 in Dummit & Foote
Let $R$ be an integral domain. Prove that if the following conditions hold then R is a Principal domain
(1) any two nonzero elements $a$ and $b$ in $R$ have a greatest common divisor which can be written in the form $ra+sb$ for some $r,s$ $\in R$, and
(2) if $a_1, a_2 , \cdots ,a_n, \cdots $ are nonzero elements of $R$ such that $a_{i+1} | a_i $, then there is a positive integer $N$ such that $a_n $ is a unit times $a_N$ for all $n \ge N$
I tried to understand the solution in
But I am stuck from the outset.
In the solution I put a link above, it claims then for any nonzero ideal $I$ in the ring $R$ of the problem, when we put partial order on $I$ as divisibility relation like
$$x \leq y \; \text{if} \; y|x $$
then each chain $C$ in $I$ with such partial order is a countable sequence $\lbrace C_i \rbrace$ with $C_{i+1} | C_i$
But why should this hold? Does divisibility partial order on (an ideal of) an integral domain make each chain be countable? This seems highly nontrivial and even seems wrong..
Any comment on the claim in the above link or on the problem itself will be a great help to me.
Thanks in advance.